(i) Which of the following is the real part of a holomorphic function. Explain your reason-...

 

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(i) Which of the following is the real part of a holomorphic function. Explain your reason- ing, but you do not need to find the holomorphic function. (a) g(x, y) = e cos(x), (b) h(x, y) = x² + y². (ii) Find a holomorphic function f: C → C that has u(x, y) = 2x³ 4x² - 6xy² + 4y² - as its real part, i.e., find a real-valued function v(x, y) such that f(z) = f(x + y) = u(x, y) + iv(x, y) for all x, y € R. (iii) Express the function f from part (ii) as a function of z E C (not as a function of x and y). (i) Which of the following is the real part of a holomorphic function. Explain your reason- ing, but you do not need to find the holomorphic function. (a) g(x, y) = e cos(x), (b) h(x, y) = x² + y². (ii) Find a holomorphic function f: C → C that has u(x, y) = 2x³ 4x² - 6xy² + 4y² - as its real part, i.e., find a real-valued function v(x, y) such that f(z) = f(x + y) = u(x, y) + iv(x, y) for all x, y € R. (iii) Express the function f from part (ii) as a function of z E C (not as a function of x and y).

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